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Properties

Label 2-819-91.30-c1-0-41
Degree $2$
Conductor $819$
Sign $-0.357 + 0.934i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.24 − 1.29i)2-s + (2.35 − 4.07i)4-s + (−1.39 − 0.806i)5-s + (2.62 + 0.292i)7-s − 6.99i·8-s − 4.17·10-s − 2.70i·11-s + (2.36 + 2.71i)13-s + (6.27 − 2.74i)14-s + (−4.34 − 7.53i)16-s + (−1.56 + 2.70i)17-s − 3.68i·19-s + (−6.56 + 3.78i)20-s + (−3.50 − 6.06i)22-s + (−0.993 − 1.71i)23-s + ⋯
L(s)  = 1  + (1.58 − 0.915i)2-s + (1.17 − 2.03i)4-s + (−0.624 − 0.360i)5-s + (0.993 + 0.110i)7-s − 2.47i·8-s − 1.31·10-s − 0.815i·11-s + (0.656 + 0.753i)13-s + (1.67 − 0.734i)14-s + (−1.08 − 1.88i)16-s + (−0.379 + 0.656i)17-s − 0.844i·19-s + (−1.46 + 0.847i)20-s + (−0.746 − 1.29i)22-s + (−0.207 − 0.358i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.357 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.357 + 0.934i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.357 + 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10850 - 3.06317i\)
\(L(\frac12)\) \(\approx\) \(2.10850 - 3.06317i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-2.62 - 0.292i)T \)
13 \( 1 + (-2.36 - 2.71i)T \)
good2 \( 1 + (-2.24 + 1.29i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.39 + 0.806i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 2.70iT - 11T^{2} \)
17 \( 1 + (1.56 - 2.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 3.68iT - 19T^{2} \)
23 \( 1 + (0.993 + 1.71i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.68 - 4.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (9.07 - 5.23i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.15 + 2.97i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.66 - 3.85i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.67 + 2.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.913 - 0.527i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.63 - 6.29i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.89 - 5.71i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 2.92T + 61T^{2} \)
67 \( 1 - 13.5iT - 67T^{2} \)
71 \( 1 + (1.17 - 0.675i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.88 + 4.55i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.10 + 5.37i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.69iT - 83T^{2} \)
89 \( 1 + (1.52 - 0.879i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.4 - 7.74i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.69415023176997652307048991743, −9.129992819774110352290132591767, −8.400405131768037956581503495850, −7.14364597725447013803728552461, −6.05033242918275804301243243756, −5.26134754369445588557763760538, −4.29843193084427355108946118310, −3.78543958186511219200568056854, −2.44793585120598909192005695031, −1.27817410419271178320519340267, 2.20177919342783015943025946910, 3.61403740804701390607589712990, 4.18168338779885285900522846386, 5.22034706620855044370179087053, 5.89501112720912114689606505901, 7.04764915483664285303250203728, 7.69864611908755572976928564130, 8.149062594657508853618634326181, 9.614684670775600278562011590757, 11.06315614548081700901931225496

Graph of the $Z$-function along the critical line